Zero-cycles over local and global fields

A classical question in number theory is whether a given system of polynomial equations with rational coefficients has a rational solution. Sets of solutions to polynomial equations give rise to what we call algebraic varieties, which are the central object of study in algebraic and arithmetic geometry. In order to answer such types of questions, one needs to detect and compute various invariants of the algebraic variety that reflect its algebraic and geometric properties. This project is concerned with the study of an invariant used for higher dimensional varieties, called the Chow group of zero-cycles, which can be used for classification of algebraic varieties, and relates to the question of existence of rational solutions to polynomial equations. On the broader impact of this award, the PI will support one graduate student and continue her various service and outreach activities including conference and seminar organization and the Bridge to the doctorate program at the University of Virginia. This project focuses on four conjectures for zero-cycles. The first conjecture concerns the construction of motivic filtrations for the Chow group of zero-cycles. The PI has prior work on this subject for abelian varieties using some K-theoretic techniques, which she now plans to extend to some special classes of K3 surfaces. The second conjecture, due to Colliot and Thelene, concerns algebraic varieties over p-adic fields, and in particular establishing this conjecture for abelian varieties using various techniques including formal group laws and integral p-adic Hodge theory. The PI also plans to study two conjectures over algebraic number fields, and the compatibility between them. The first of these is a conjecture of Kato and Saito, which can be thought of as a local-to-global principle for zero-cycles, and she will explore whether her work could constitute the first step towards a new type of Euler system. The second conjecture is part of the famous Beilinson-Bloch conjectures, which predict that the Chow group of zero-cycles is a finitely generated abelian group. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.